← 2020 Paper 1

UPSC 2020 Maths Optional Paper 1 Q2c — Step-by-Step Solution

15 marks · Section A

Cylinder · Analytic Geometry · asked 4× in 13 yrs · Read the full method →

Question

Find the equation of the cylinder whose generators are parallel to the line $\frac{x}{1}=\frac{y}{-2}=\frac{z}{3}$ and whose guiding curve is $x^2+y^2=4,\ z=2$.

Technique

Parametrize the generator through a general point, intersect with the guiding-curve plane, substitute into the guiding curve, then relabel.

Solution

A cylinder is the union of straight lines (generators) all parallel to a fixed direction $\mathbf{d}=(1,-2,3)$, each meeting the guiding curve $\Gamma:\ x^2+y^2=4,\ z=2$.

Step 1 — Generator through a general point

Let $P=(\alpha,\beta,\gamma)$ be any point on the cylinder. The generator through $P$ is

Step 2 — Intersect the generator with the plane $z=2$ of the guiding curve

Set $\gamma+3t=2\Rightarrow t=\dfrac{2-\gamma}{3}$. The point where the generator meets $z=2$ is

Step 3 — Impose that this point lies on the guiding curve

$\Gamma$ requires $x_0^2+y_0^2=4$:

Multiply by $9$:

Drop the subscripts (replace $(\alpha,\beta,\gamma)\to(x,y,z)$) to get the locus of all such $P$:

Answer